A resonator is a device which passes only a single frequency or a small band of frequencies out of a large band of frequencies submitted to it. A measurement of the suitability of a material as a crystal resonator is its quality factor, Q, which is equal to f.sub.o /.DELTA.f where f.sub.o is the center frequency passed by the resonator and .DELTA.f is the 3db bandwidth passed. Only piezoelectric resonators have the high Q values and small size required for most modern electronic devices.
In addition to a high Q value, a piezoelectric resonator should be physically stable and should pass the same frequency as it ages (time stability). Another very important property is the ability of the resonator to pass the same frequency in spite of changes in its temperature. So far, the best material appears to be quartz, which is the most widely used material for piezoelectric resonators.
Quartz has excellent physical stability and a high Q value (up to 10.sup.6). Moreover, quartz is unique in that over a limited temperature range the frequency of the passed signal stays almost constant. However, quartz has another property which is very undesirable, a high C.sub.0 /C.sub.1 ratio of about 250.
Referring to FIG. 1, an electrical schematic, the drawing shows the electrical equivalent of a crystal resonator. In the drawing a capacitor C.sub.1, an inductance L.sub.1, and a resistance r.sub.1 in series are connected in parallel to a capacitance C.sub.0. Ideally C.sub.0 (and r.sub.1) would be zero, but in practical crystal resonators C.sub.0 is never zero and therefore the ratio C.sub.0 /C.sub.1 is not zero. The C.sub.0 /C.sub.1 ratio is related to the physical properties of the material used in the crystal resonator according to the formula C.sub.0 /C.sub.1 = (.pi.kq.sub.o)/(32.epsilon..sup.2) where k is the effective dielectric constant, q.sub.o is the stiffness, and .epsilon. is the piezoelectric constant. The value of C.sub.0 is determined by the formula C.sub.0 = .epsilon..sub.o .epsilon..sub.R A/T where .epsilon..sub.0 is the permeativity of free space, .epsilon..sub.R is the relative dielectric constant of the crystal, A is the area of the electrode, and T is the crystal thickness. The consequences of a large C.sub.0 /C.sub.1 ratio are shown in FIG. 2.
FIG. 2 is a graph showing the relationship between the amplitude of the electrical signal passed at various frequencies by a crystal resonator. The solid line shows the amplitudes of the signals for a crystal resonator with a large C.sub.0 /C.sub.1 ratio such as a quartz resonator and the dotted line shows the amplitudes for a crystal resonator with a small C.sub.0 /C.sub.1 ratio. The drawing indicates that for a material such as quartz an anti-resonance occurs at a frequency, f.sub.1, very near f.sub.o, the center frequency, and at f.sub.1 no signal is passed. The separation, .DELTA.f.sub.p, is equal to 1/2 f.sub.o C.sub.1 /C.sub.0 and therefore materials with a low C.sub.0 /C.sub.1 ratio have a greater .DELTA.f.sub.p as shown by the dotted line.
The result of using crystal resonators having a large C.sub.0 /C.sub.1 ratios in a filter is shown in FIG. 3, a graph of amplitude against frequency for a wideband filter, such as one made from quartz resonators. A wideband filter contains crystal resonators in the circuit, but passes a wider range of frequencies. FIG. 3 shows that if the C.sub.0 /C.sub.1 ratio is large there will be a frequency f.sub.1 in the middle of the band which is not passed. If a material is used in the filter which has a small C.sub.0 /C.sub.1 ratio the gap in the middle of the band vanishes. (Efforts have been made to find other suitable resonator materials which have a low C.sub.0 /C.sub.1 ratio. For example, both piezoceramic (barium titanate and related materials), and lithium tantalate have lower C.sub.0 C.sub.1 ratios, (about 25 for piezoceramic and about 20 for LiTaO.sub.3) but piezoceramic has poor time and temperature stability and lithium tantalate has poor physical and temperature stability.)
Finally, while narrow band filters can be made from quartz resonators by simply connecting the resonators in series and coupling each resonator to a capacitor, wideband filters made from quartz resonators require considerably more complicated circuitry. Wideband resonators are needed for FM radios, high frequency TV, IF (intermediate frequency) transmission, radar, and pulse transmission circuits, and a simpler circuitry would be less expensive.